This article presents new rationality results for the ratios of critical values of Rankin–Selberg L-functions of $\mathrm {GL}(n) \times \mathrm {GL}(n')$ over a totally imaginary field $F.$ The proof is based on a cohomological interpretation of Langlands’s contant term theorem via rank-one Eisenstein cohomology for the group $\mathrm {GL}(N)/F,$ where $N = n+n'.$ The internal structure of the totally imaginary base field has a delicate effect on the Galois equivariance properties of the critical values.

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$-grading that can be parametrized by a quadrangular algebra.

One of our key tools is a new definition of exponential maps that makes sense even over fields of characteristic $2$ and $3$, which ought to be interesting in its own right.

Not only was Jacques Tits a constant source of inspiration through his work, he also had a direct personal influence, notably through his threat to speak evil of our work if it did not include the characteristic 2 case.

—The Book of Involutions [KMRT98, p. xv]

We introduce a new algebra $\mathcal {U}=\dot {\mathrm {\mathbf{U}}}_{0,N}(L\mathfrak {sl}_n)$ called the shifted $0$-affine algebra, which emerges naturally from studying coherent sheaves on n-step partial flag varieties through natural correspondences. This algebra $\mathcal {U}$ has a similar presentation to the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk. Then, we construct a categorical $\mathcal {U}$-action on a certain 2-category arising from derived categories of coherent sheaves on n-step partial flag varieties. As an application, we construct a categorical action of the affine $0$-Hecke algebra on the bounded derived category of coherent sheaves on the full flag variety.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

In an earlier work, we defined a “generalised Temperley–Lieb algebra” $TL_{r, 1, n}$ corresponding to the imprimitive reflection group G(r, 1, n) as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley–Lieb algebra $TL_{r, p, n}$ which corresponds to the complex reflection group G(r, p, n). Our definition identifies $TL_{r, p, n}$ as the fixed-point subalgebra of $TL_{r, 1, n}$ under a certain automorphism $\sigma$. We prove the cellularity of $TL_{r, p, n}$ by proving that $\sigma$ induces a special shift automorphism with respect to the cellular structure of $TL_{r, 1, n}$. We also give a description of the cell modules of $TL_{r, p, n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

We extend a comparison theorem of Anandavardhanan–Borisagar between the quotient of the induction of a mod $p$ character by the image of an Iwahori–Hecke operator and compact induction of a weight to the case of the trivial character. This involves studying the corresponding non-commutative Iwahori–Hecke algebra. We use this to give an Iwahori theoretic reformulation of the (semi-simple) mod $p$ local Langlands correspondence discovered by Breuil and reformulated functorially by Colmez. This version of the correspondence is expected to have applications to computing the mod $p$ reductions of semi-stable Galois representations.

We determine the cohomology of the closed Drinfeld stratum of p-adic Deligne–Lusztig schemes of Coxeter type attached to arbitrary inner forms of unramified groups over a local non-archimedean field. We prove that the corresponding torus weight spaces are supported in exactly one cohomological degree and are pairwise non-isomorphic irreducible representations of the pro-unipotent radical of the corresponding parahoric subgroup. We also prove that all Moy–Prasad quotients of this stratum are maximal varieties, and we investigate the relation between the resulting representations and Kirillov’s orbit method.

We investigate properties of closed approximate subgroups of locally compact groups, with a particular interest for approximate lattices (i.e., those approximate subgroups that are discrete and have finite co-volume).

We prove an approximate subgroup version of Cartan’s closed-subgroup theorem and study some applications. We give a structure theorem for closed approximate subgroups of amenable groups in the spirit of the Breuillard–Green–Tao theorem. We then prove two results concerning approximate lattices: we extend to amenable groups a structure theorem for mathematical quasi-crystals due to Meyer; we prove results concerning intersections of radicals of Lie groups and discrete approximate subgroups generalising theorems due to Auslander, Bieberbach and Mostow. As an underlying theme, we exploit the notion of good models of approximate subgroups that stems from the work of Hrushovski, and Breuillard, Green and Tao. We show how one can draw information about a given approximate subgroup from a good model, when it exists.

In this paper, we compare the $\mathbb J$-stratification (or the semi-module stratification) and the Ekedahl–Oort stratification of affine Deligne–Lusztig varieties in the superbasic case. In particular, we classify the cases where the $\mathbb J$-stratification gives a refinement of the Ekedahl–Oort stratification, which include many interesting cases such that the affine Deligne-Lusztig variety admits a simple geometric structure.

We define oriented Temperley–Lieb algebras for Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan–Lusztig polynomials for these spaces.

Let be a commutative ring containing a primitive $l'$th root $\varepsilon $ of $1$. The infinitesimal q-Schur algebras over form an ascending chain of subalgebras of the q-Schur algebra , which are useful in studying representations of the Frobenius kernel of the associated quantum linear group. Let be the quantized enveloping algebra of $\mathfrak {gl}_n$ over . There is a natural surjective algebra homomorphism $\zeta _{d}$ from to . The map $\zeta _{d}$ restricts to a surjective algebra homomorphism $\zeta _{d,r}$ from to , where is a certain Hopf subalgebra of , which is closely related to Frobenius–Lusztig kernels of . We give the extra defining relations needed to define the infinitesimal q-Schur algebra as a quotient of . The map $\zeta _{d,r}$ induces a surjective algebra homomorphism , where is the modified quantum algebra associated with . We also give a generating set for the kernel of $\dot {\zeta }_{d,r}$. These results can be used to give a classification of irreducible -modules over a field of characteristic p.

Let $\mathbf {G}$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk $ and ${\mathbf B}$ be a Borel subgroup of ${\mathbf G}$. In this paper, we completely determine the composition factors of the permutation module $\mathbb {F}[{\mathbf G}/{\mathbf B}]$ for any field $\mathbb {F}$.

We give a precise classification, in terms of Shimura data, of all $1$-dimensional Shimura subvarieties of a moduli space of polarized abelian varieties.

Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree $\displaystyle |Syl_r(G)|$ is said to be weak r-Steinberg if it vanishes on the r-singular elements of $G.$ In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group $GL(n,q).$ Then we characterize the quasi r-Steinberg characters of $GL(2,q)$ and $GL(3,q).$ Finally, we obtain a classification of the weak r-Steinberg characters of $GL(n,q).$

In this paper, we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings R of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system $\Phi $ of rank $\ge 2$, there exists a universal bound $L=L(\Phi )$ such that the simply connected Chevalley groups $G(\Phi ,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type R.

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. Among the applications are precise connections between Ringel duality, split quasi-hereditary covers and double centralizer properties, constructions of split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of q-Schur algebras and a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring. They are shown to behave nicely under change of rings techniques.

In this paper, the authors introduce a new notion called the quantum wreath product, which is the algebra $B \wr _Q \mathcal {H}(d)$ produced from a given algebra B, a positive integer d and a choice $Q=(R,S,\rho ,\sigma )$ of parameters. Important examples that arise from our construction include many variants of the Hecke algebras, such as the Ariki–Koike algebras, the affine Hecke algebras and their degenerate version, Wan–Wang’s wreath Hecke algebras, Rosso–Savage’s (affine) Frobenius Hecke algebras, Kleshchev–Muth’s affine zigzag algebras and the Hu algebra that quantizes the wreath product $\Sigma _m \wr \Sigma _2$ between symmetric groups.

In the first part of the paper, the authors develop a structure theory for the quantum wreath products. Necessary and sufficient conditions for these algebras to afford a basis of suitable size are obtained. Furthermore, a Schur–Weyl duality is established via a splitting lemma and mild assumptions on the base algebra B. Our uniform approach encompasses many known results which were proved in a case by case manner. The second part of the paper involves the problem of constructing natural subalgebras of Hecke algebras that arise from wreath products. Moreover, a bar-invariant basis of the Hu algebra via an explicit formula for its extra generator is also described.

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$. In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$-valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$-valued representations.

Given a connected reductive algebraic group G over an algebraically closed field, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.

We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$-hard, and computing them is ${{{\textsf {#P}}}}$-hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.